Nicola Ciccoli

Noncommutative Instantons on the 4-Sphere¶from Quantum Groups

Abstract: We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle ?7→?4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson–Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SUq(2); it determines a new deformation of the 4-sphere ∑4q as the algebra of coinvariants in ?q7. We show that the quantum vector bundle associated to the fundamental corepresentation of SUq(2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑4q, we define two 0-summable Fredholm modules and we compute the Chern–Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non-trivial.

Bijectivity of the canonical map for the non-commutative instanton bundle

Abstract It is shown that the quantum instanton bundle introduced in [Commun. Math. Phys. 226 (2002) 419] has a bijective canonical map and is, therefore, a coalgebra Galois extension.

The quantization of the symplectic groupoid of the standard Podle s ̀ sphere

Abstract We give an explicit form of the symplectic groupoid G ( S 2 , π ) that integrates the semiclassical standard Podles sphere ( S 2 , π ). We show that Sheu’s groupoid G S , whose convolution C ∗ -algebra quantizes the sphere, appears as the groupoid of the Bohr–Sommerfeld leaves of a (singular) real polarization of G ( S 2 , π ) . By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.

Twisted Reality Condition for Dirac Operators

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones, we propose a new twisted reality condition for the Dirac operator.

On the integration of Poisson homogeneous spaces

Abstract We study a reduction procedure for describing the symplectic groupoid of a Poisson homogeneous space obtained by quotient of a coisotropic subgroup. We perform it as a reduction of the Lu–Weinstein symplectic groupoid integrating Poisson Lie groups, that is suitable even for the non-complete case.

Quantum Even Spheres Σ2nq from Poisson Double Suspension

We define even dimensional quantum spheres Σ2nq that generalize to higher dimension the standard quantum two-sphere of Podleś and the four-sphere Σ4q obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podleś two-sphere. The Poisson spheres that we get have the same kind of symplectic foliation consisting of a degenerate point and a symplectic ℝ2n and, after quantization, have the same C*–algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.

Induction of quantum group representations

Abstract In this paper we will define a generalized procedure of induction of quantum group representations both from quantum and from coisotropic subgroups proving also their main properties. We will then show that such a procedure realizes quantum group representations on generalized quantum bundles.

Quantization of projective homogeneous spaces and duality principle

We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.

The coisotropic subgroup structure of SLq(2,R)

We study the coisotropic subgroup structure of standard SL_q(2,R) and the corresponding embeddable quantum homogeneous spaces. While the subgroups S^1 and R_+ survive undeformed in the quantization as coalgebras, we show that R is deformed to a family of quantum coisotropic subgroups whose coalgebra can not be extended to an Hopf algebra. We explicitly describe the quantum homogeneous spaces and their double cosets.

Quantum even spheres Sigma_q^2n from Poisson double suspension

We define even dimensional quantum spheres Σ2nq that generalize to higher dimension the standard quantum two-sphere of Podleś and the four-sphere Σ4q obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podleś two-sphere. The Poisson spheres that we get have the same kind of symplectic foliation consisting of a degenerate point and a symplectic ℝ2n and, after quantization, have the same C*–algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.